Finally, we use DGSPT in order to analyse fast-slow Poincaré maps associated to fast-slow systems in standard with a single slow variable, a situation which arises often in applications if one allows a parameter \(\alpha \)  to evolve slowly in time; see e.g. [1]}, [2]} and the references therein for examples in the context of dynamic bifurcation theory. On the assumption that the (continuous-time) layer problem has a hyperbolic limit cycle for \(\alpha = \alpha _\ast \) , there exists a 2-dimensional manifold of limit cycles \(\mathcal {M}\)  in \(\mathbb {R}^n \times V_\alpha \) , where \(V_\alpha \)  is a sufficiently small neighbourhood about \(\alpha _\ast \)  (see e.g. [3]} for an example in 3 dimensions). After showing that the Poincaré map on a transverse section \(\Delta \)  is a fast-slow map with normally hyperbolic critical manifold \(S = \Delta \cap \mathcal {M}\) , we are able to characterise the geometry and dynamics of the Poincaré map in detail using DGSPT. Using information about the Poincaré map we can then infer geometric and dynamical properties for the higher-dimensional ODE system, including the persistence of \(\mathcal {M}\)  a nearby locally invariant manifold \(\mathcal {M}_\varepsilon \) .
